International Financial Management 5

Figure 7.2: IRPT Line id -if  B (Capital inflows) - Example 1 IRPT Line  A (Capital outflows) - Example 2 p (forward premium) Consider a point under the IRPT line, say A (like in Example 2): p > id -if (or p + if > id)  a long spot/short forward position has a higher yield than borrowing abroad at if and investing at home at id  Arbitrage is possible! (There is a pricing mistake). Covered Arbitrage Strategy: (1) Borrow DC at id for T days (2) Convert DC to FC (the long position in FC) (3) Deposit FC at if for T days (4) Sell forward FC at Ft,T (the short forward position) That is, today, at a point like A, domestic capital fly to the foreign country: What an investor pays in domestic interest rate, id, is more than compensated by the high forward premium, p, and the foreign interest rate, if. • IRPT: Evidence Starting from Frenkel and Levich (1975), there is a lot of evidence that supports IRPT. For example, Figure 7.3 plots the daily interest rate differential against the annualized forward premium. They plot very much along the 45° line. Moreover, the correlation coefficient between these two series is 0.995, highly correlated series! IFM-LN.62 Figure 7.3: IRPT Line – USD/GBP (daily, 1990-2015) 0.02 0.00 ‐0.08 ‐0.06 ‐0.04 ‐0.02 0.00 0.02 ‐0.02 i_USD‐i_GBP ‐0.10 ‐0.04 ‐0.06 ‐0.08 ‐0.10 premium_1Y Using intra-daily data (10’ intervals), Taylor (1989) also find strong support for IRPT. At the tickby-tick data, Akram, Rice and Sarno (2008, 2009) show that there are short-lived (from 30 seconds up to 4 minutes) departures from IRP, with a potential profit range of 0.0002-0.0006 per unit. The short-lived nature and small profit range point out to a fairly efficient market, with the data close to the IRPT line. There are situations, however, where we observe significant and more persistent deviations from the IRPT line. These situations are usually attributed to monetary policy, credit risk, funding conditions, risk aversion of investors, lack of capital mobility, default risk, country risk, and market microstructure effects. For example, during the 2008-2009 financial crisis there were several violations of IRPT (in Graph 7.2, the point well over the line (-.0154,-.0005) is from May 2009). These violations are attributed to funding constraints –i.e., difficulties to do step (1): borrow. See Baba and Parker (2009) and Griffoli and Ranaldo (2011). Chapter 7 - Appendix – Taylor Series Definition: Taylor Series Suppose f is an infinitely often differentiable function on a set D and c ∈D. Then, the series Tf(x, c) = Σn [ f(n)(c)/n!] (x - c)n is called the (formal) Taylor series of f centered at, or around, c. Note: If c=0, the series is also called MacLaurin Series. Taylor Series Theorem Suppose f ∈ Cn+1([a, b]) -i.e., f is (n+1)-times continuously differentiable on [a, b]. Then, for c ∈ IFM-LN.63 [a,b] we have: f ( x )  T ( x, c )  R  where Rn 1 ( x)  / // f (n)  x 0  f c  x  c 0  f c  x  c 1  f c  x  c 2     x  x 0 n  R 0! 1! 2! n! 1 n f (n1 )  p x  p  dp  n! In particular, the Tf(x, c) for an infinitely often differentiable function f converges to f iff the remainder R(n+1)(x) →converges to 0 as n → ∞. Example: 1st-order Taylor series expansion, around c=0, of f(x) = log(1+xd) , where d is a constant f(x) = log(1+x d) f’ (x0=0) = 0 f’ (x) = d/(1+x d) f ’(x0=1) = d  1st-order Taylor’s series formula (n=1): log(1+x d) T(x; c) = 0 + d (x-0) = x d  if d=1, then log(1+x) Application: IRP Approximation Ft ,T  S t We start with IRP: Now, take logs: (1  id xT / 360) (1  i f xT / 360) log(Ft ,T )  log(S t )  log(1  id xT / 360)  log(1  i f xT / 360) After simple algebra: log(Ft ,T )  log(S t )  log(1  id xT / 360)  log(1  i f xT / 360) Recall that log changes can approximate percentage changes: Then, using the approximation for log(1 + xd) = x d, we get Solving for Ft,T, gets the linearized approximation to IRP. IFM-LN.64 Ft ,T  S t log(Ft ,T )  log(S t )  St Ft ,T  S t  i d xT / 360  i f xT / 360 St x CHAPTER 7 – BRIEF ASSESMENT 1. Assume the following information: St.USD/AUD = .80 USD/AUD St,USD/GBP = 1.40 USD/GBP St,AUD/GBP = 1.80 AUD/GBP Is triangular arbitrage possible? If so, explain the steps that would reflect triangular arbitrage, and compute the profit from this strategy (expressed as a % per unit borrowed). Explain how market forces move to eliminate triangular arbitrage’s profits. 2. Difficult. Let’s complicate triangular arbitrage, by introducing bid-ask spreads. Assume the following information: St,USD/AUD = .81-.82 USD/AUD St,USD/GBP = 1.40-1.42 USD/GBP Calculate an arbitrage-free cross rate (AUD/GBP) quote (with bid-ask spread). 3. Assume the following information: St = 1.10 USD/EUR iEUR = 1.50% iUSD = 2.75% T = 180 days (A) Determine the arbitrage-free 180-day forward rate (use IRP). (B) Suppose Bank Q offers FQt,180 = 1.12 USD/EUR. Given this information, is covered interest arbitrage possible? Design a covered arbitrage strategy and calculate its profits. (C) Suppose Bank P offers FPt,180 = 1.08 USD/EUR. Given this information, is covered interest arbitrage possible? Design a covered arbitrage strategy and calculate its profits. 4. Assume the following information: St = 1.40 USD/GBP FQt,270 = 1.42 USD/GBP iGBP = 2.50% iUSD = 2.75% T = 180 days Calculate p (the forward premium) and the interest rate differential. What kind of capital flows the U.K. economy will experience? IFM-LN.65 CHAPTER 7 - BONUS COVERAGE: IRPT with Bid-Ask Spreads Exchange rates and interest rates are quoted with bid-ask spreads. Consider a trader in the interbank market: She will have to buy or borrow at the other party's ask price. She will sell or lend at the bid price. There are two roads to take for arbitrageurs: borrow domestic currency or borrow foreign currency. • Bid’s Bound: Borrow Domestic Currency (1) A trader borrows DC 1 at time t=0, and repays 1+iask,d at time=T. (2) Using the borrowed DC 1, she can buy spot FC at (1/Sask,t). (3) She deposits the FC at the foreign interest rate, ibid,f. (4) She sells the FC forward for T days at Fbid,t,T This strategy would yield, in terms of DC: (1/Sask,t) (1+ibid,f) Fbid,t,T. In equilibrium, this strategy should yield no profit. That is, (1/Sask,t) (1+ibid,f) Fbid,t,T  (1+iask,d). Solving for Fbid,t,T, Fbid,t,T  Sask,t * , 𝑇 ∗ 360 𝑇 1 𝑖𝑏𝑖𝑑,𝑓 ∗ 360 = Ubid. • Ask’s Bound: Borrow Foreign Currency (1) The trader borrows FC 1 at time t=0, and repay 1+iask,f. (2) Using the borrowed FC 1, she can buy spot DC at Sask,t. (3) She deposits the DC at the foreign interest rate, ibid,d. (4) She buys the FC forward for T days at Fask,t,T Following a similar procedure as the one detailed above, we get: Fask,t,T  Sbid,t ∗ 𝑇 1 𝑖𝑏𝑖𝑑,𝑑 ∗ 360 𝑇 1 𝑖𝑎𝑠𝑘,𝑓 ∗ 360 = Lask. Together with Fask,t,T > Fbid,t,T, these bounds are used by traders to establish arbitrage-free quotes. Figure 7.4 illustrates the bounds. IFM-LN.66 Figure 7.4: Trading bounds for the Forward bid and the Forward ask. Fask,t,T Fbid,t,T Lask Ubid Ft,T Example: IRPT bounds at work. Data: St = 1.6540 – 1.6620 USD/GBP iUSD = 7¼-½, iGBP = 8 1/8 – 3/8, Ft,one-year = 1.6400 – 1.6450 USD/GBP. Check if there is an arbitrage opportunity (we need to check the bid’s bound and ask’s bound). i) Bid’s bound covered arbitrage strategy: 1) Borrow USD 1 at 7.50% for 1 year  we will repay USD 1.07500 at T=1 year  we get GBP 1/1.6620 = GBP 0.6017 2) Convert to GBP 3) Deposit GBP 0.6017 at 8.25% 4) Sell GBP forward at 1.64 USD/GBP  we get (1/1.6620) x (1 + .08125)x1.64 = USD 1.06694  No arbitrage opportunity. For each USD we borrow, we lose USD .00806. ii) Ask’s bound covered arbitrage strategy: 1) Borrow GBP 1 at 8.375% for 1 year  we will repay GBP 1.08375 at T=1 year  we get USD 1.6540 2) Convert to USD 3) Deposit USD 1.6540 at 7.250% 4) Buy GBP forward at 1.645 USD/GBP  we get 1.6540x(1 + .07250)x(1/1.6450) = GBP 1.07837  No arbitrage opportunity. For each GBP we borrow, we lose GBP 0.0054. Note: The bid-ask forward quote is consistent with no arbitrage. That is, the forward quote is within the IRPT bounds. Check: Ubid = Sask,t * , , . = 1.6620 USD/GBP * = 1.6524 USD/GBP .  Ubid  Fbid,t,T = 1.6400 USD/GBP. Lask = Sbid,t * , , = 1.6540 USD/GBP * . . = 1.6368 USD/GBP  Lask  Fask,t,T = 1.6450 USD/GBP. ¶ IFM-LN.67 Chapter 8 – Theories of FX Determination – Part 1 In Chapter 4, we briefly mentioned two theories of exchange rate determination: The Balance of Trade (BT) Approach and the Monetary Approach (MA) Under the BT Approach, net trade flows (X-M) are the main determinants of St. According to this approach, we expect an increase (decrease) in the TB to depreciate (appreciate) the FC. That is, 𝑒 , = (St/St-1) – 1 = f(TBt), where f’<0. Under the MA, St is determined by the relative money demand and money supply between the two currencies: 𝑆 = f(Ld,T / Lf,T , MSd,T / MSf,T,,...), where f1<0 & f2>0. In this chapter, we develop more theories to explain St. The emphasis will be on arbitrage, actually pseudo-arbitrage, theories, focusing on equilibrium in only one market. That is, we will rely on partial equilibrium stories to explain St. Our goal is to find an explicit functional form for 𝑆 , say 𝑆 = α+β Xt, where Xt. is a variable or set of variables determined by a theory. Different theories will have different Xt and or different f(.). Eventually, we would like to have a precise mathematical formula to forecast St+T. Q: How do we know the formula of 𝑆 is any good? • Testing a Theory We will judge a theory by how well it explains the behavior of the observed 𝑆 . For example, a good theory should match the observed behavior of the quarterly MXN/USD exchange rate for the 1978 - 2022 period, as shown in Figure 8.1. IFM-LN.68 Figure 8.1: Behavior of the MXN/USD (1978-2022) Like many macroeconomic series, exchange rates have a trend, see Figure 8.1 above –in statistics, these trends in macroeconomic series are called stochastic trends. It is better to work with changes, not levels. As can be seen in Figure 8.2, the trend is gone after calculating changes in 𝑆 . Figure 8.2: Behavior of the Changes in MXN/USD (1993-2022) ow, the trend, which in many cases is easy to explain, is gone. Our goal will be to explain ef,t, the percentage change in 𝑆 . Goal: 𝑆 = f( id, if,Id, If,…). But, it will be easier to explain ef,t= (St – St-1)/St-1 = f( id, if, Id, If,…). Once we get ef,t, we get 𝑆  𝑆 = St-1 x (1 + ef,t) The 𝑆 that we will obtain will be an equilibrium value. That is, the 𝑆 we will be calculated using a model that assumes some kind of equilibrium in the FX market. IFM-LN.69 Q: How are we going to test our equilibrium values? A: We would like our theory to match the data, say the mean and standard deviation of St. Figure 8.3 plots the distribution of ef.(MXN/USD); calculated from monthly data during 1987-2017. Below Figure 8.3, we show the descriptive statistics for the distribution of ef.(MXN/USD). Figure 8.3: Distribution of the Changes in MXN/USD (1993-2022) Descriptive Stats for ef (MXN/USD): ef (USD/MXN) Mean Standard Error Median Standard Deviation Sample Variance Kurtosis Skewness Minimum Maximum Count 0.0052 0.0019 0.0004 0.0351 0.0021 33.3631 3.9122 -0.0887 0.3500 350 The usual (average, expected) monthly percentage change represents a 0.52% appreciation of the USD against the MXN (annualized change: 6.24%). The standard deviation is 3.51%. The mean, standard deviation, skewness and kurtosis are called unconditional moments. Theories also produce conditional moments -i.e., conditional on the theory. In general, we associate matching unconditional moments with long-run features of a model; while we associate matching conditional moments with short-run features of a model. • Review from Chapter 7 Effect of arbitrage on FX Markets: Local arbitrage  sets consistent rates across banks IFM-LN.70 Triangular arbitrage Covered arbitrage  sets cross rates  sets a relation between Ft,T, St ,id, if (IRPT)  Ft,T = St (1 + id * T/360)/(1 + if * T/360). In this class, we will study the effect of “arbitrage” in goods (PPP) and financial flows (IFE) on FX Markets. We will generate explicit models for St. 8.1. Purchasing Power Parity (PPP) PPP is based on the law of one price (LOOP): same goods once denominated in a common currency should have the same price. If they are not, then pseudo-arbitrage is possible. Example: LOOP for Oil. Poil-USA = USD 40. Poil-SWIT = CHF 80. LOOP: Poil-SWIT 𝐒𝐭𝐋𝐎𝐎𝐏 = Poil-USA  𝐒𝐭𝐋𝐎𝐎𝐏 = Poil-USA/Poil-SWIT =USD 40/CHF 80 =0.50 USD/CHF. Suppose St = 0.75 USD/CHF  Poil-SWIT (in USD) = CHF 80 x 0.75 USD/CHF = USD 60. That is, a barrel of oil in Switzerland is more expensive -once denominated in USD- than in the US. Arbitrageurs/traders will buy oil in the U.S. (to export it to Switzerland) and simultaneously sell oil in Switzerland. This movement of oil will simultaneously increase the price of oil in the U.S. (Poil-USA ↑); decrease the price of oil in Switzerland (Poil-SWIT ↓); and appreciate the USD against the CHF (St ↓). ¶ LOOP Notes: ⋄ LOOP gives an equilibrium exchange rate (EER, in the econ lit). Equilibrium will be reached when there is no trade in oil (because of pricing mistakes). That is, when the LOOP holds for oil. ⋄ LOOP is telling us what St should be (in equilibrium): StLOOP. It is not telling what St is in the market. It is just an implied rate from market prices. ⋄ Using the LOOP, we have generated a model for St. (Recall that a model is an attempt to explain and predict economic phenomena.) When applied to a price index, we will call this model, the PPP model. ⋄ The generated model, like all models, is a simplification of the real world. For example, we have ignored (or implicitly assumed negligible) trade frictions (transportation costs, tariffs, etc.). Problem for the LOOP: There are many traded goods in the economy. Solution: Work with baskets of goods that represent many goods. For example, the CPI basket (in the U.S., we use the CPI-U, which reflects spending patterns for urban consumers), which includes housing (41%), transportation (17%), food & beverages (15%), health care (7%), recreation (6%), etc. The price of a basket is the weighted average price of the components. For example: Price CPI-U basket = .41 * Price of housing + .17 * Price of transportation + ... The price of the CPI basket is usually referred as the “price level” of an economy. 8.1.1 Absolute version of PPP IFM-LN.71 The FX rate between two currencies is simply the ratio of the two countries' general price levels: 𝐒𝐭𝐏𝐏𝐏 = Domestic Price level / Foreign Price level = Pd / Pf (Absolute PPP) Example: Law of one price for CPIs. CPI-basketUSA = USD 755.3 CPI-basketSWIT = CHF 1241.2  StPPP = USD 755.3/CHF 1241.2 = 0.6085 USD/CHF. If St  0.6085 USD/CHF, there will be trade of the goods in the basket between Switzerland and US. Suppose St = 0.70 USD/CHF > 𝐒𝐭𝐏𝐏𝐏 = 0.6085 USD/CHF. Then CPI-basketSWIT (in USD) = CHF 1241.2 * 0.70 USD/CHF = USD 868.70 > CPI-basketUSA “Things” –i.e., the components in the CPI basket- are, on average, cheaper in the U.S. There is a potential profit from trading the CPI basket’s components: Potential profit: USD 868.70 – USD 755.3 = USD 93.40 Traders will do the following “pseudo-arbitrage” strategy: 1) Borrow USD 2) Buy the CPI-basket in the US 3) Sell the CPI-basket, purchase in the US, in Switzerland. 4) Sell CHF/Buy USD 5) Repay the USD loan, keep the profits. (CPI-basketUSA ↑) (CPI-basketSWIT ↓)  StPPP ↑ (St (USD/CHF) ↓) Note: Prices move and push St (market price) & 𝐒𝐭𝐏𝐏𝐏 (equilibrium price) towards convergence. ¶ Under PPP, a USD buys the same amount of goods in the U.S. and in Switzerland. That is, a USD has the same purchasing power in the U.S. & in Switzerland. Vice versa, a CHF buys the same amount of goods in Switzerland and in the U.S. • Absolute PPP: The Real Exchange Rate The absolute version of the PPP theory is expressed in terms of St, the nominal exchange rate. We can express the absolute version of the PPP relationship in terms of the real exchange rate, Rt. That is, Rt = 𝑆 Pf / Pd. The real exchange rate allows us to compare foreign prices, translated into domestic terms, with domestic prices. It is common to associate Rt > 1 with a more efficient/productive domestic economy. If absolute PPP holds  Rt = 1. Terminology: If Rt ↑, foreign goods become more expensive relative to domestic goods. We say there is “a real depreciation of the DC”. Similarly, if Rt ↓, we say there is “a real appreciation of the DC.” IFM-LN.72 Example: Suppose a basket –the Big Mac (sesame-seed bun, onions, pickles, cheese, lettuce, beef patty and special sauce)– costs CHF 6.50 and USD 4.93 in Switzerland and in the U.S., respectively. Pf = CHF 6.50 Pd = USD 4.93 St = 0.9909 USD/CHF. Rt = St PSWIT / PUS = 0.9908 USD/CHF * CHF 6.50/USD 4.93 = 1.3065. Taking the Big Mac as our basket, the U.S. is more competitive than Switzerland. Swiss prices are higher (Rt - 1 = 30.7% higher!) than U.S. prices, after taking into account the nominal exchange rate. That is, with one USD, we consume 30.7% more in the U.S. than in Switzerland. To bring the economy back to equilibrium –no trade on Big Macs-, we expect the USD to appreciate against the CHF. According to PPP, the USD is undervalued against the CHF:  Trading signal: Buy USD/Sell CHF. Note: Obviously, we do not expect to see Swiss consumers importing Big Macs from the U.S.; but the components of the Big Mac are internationally traded. Trade would happen in the components! ¶ Indicator of under/over-valuation: Rt > 1  FC is overvalued. Note: In the short-run, we will not take our cars to Mexico to be repaired, because a mechanic’s hour is cheaper than in the U.S. But in the long-run, resources (capital, labor) will move, likely to produce cars in Mexico to export them to the U.S. We can think of the over-/under-valuation as an indicator of movement of resources. Remark: If St changes, but Pf & Pd move in such a way that Rt remains constant, changes in St do not affect firms. There is no change in real cash flows. Aside: Economists like to work with logs. Thus, using lower case letters to denote the log of variables, x=log(X), we can rewrite the real exchange rate as: rt = st + pf,t – pd,t If PPP holds, then rt = 0. Then, st = pf,t – pd,t. • Absolute PPP: Real v. Nominal Exchange Rates Economists think that monetary variables affect nominal variables, like prices and the nominal exchange rate, St. But, monetary variables do not affect real variables. In this case, only relative demands and supplies affect Rt. For example, an increase in U.S. output relative to European output (say, because of a technological innovation) will decrease PUS relative to PEUR  Rt ↑ (a real depreciation of the USD). On the other hand, a monetary approach to exchange rates, predicts that an increase in the U.S. money supply will increase PUS and, thus, an increase in St, but no effect on Rt. IFM-LN.73 • Absolute PPP: Does it hold? We use a basket of goods to test PPP. To get better results, it is a good idea to use the same basket (or comparable baskets). For example, the Big Mac. Example: The Economist’s Big Mac Index, shown in Exhibit 8.1, shows the over/undervaluation of a currency relative to the USD. That is, it shows the real exchange rate, Rt - 1. Rt = St PBigMac,d / PBigMac,d,f Test: If Absolute PPP holds  Rt = 1 (& over/undervaluation=0!). Exhibit 8.1: The Economist’s Big Mac Index (July 2019) IFM-LN.74 IFM-LN.75 Check: http://www.economist.com/content/big-mac-index There are big deviations from Absolute PPP, which can vary a lot over time. See Figure 8.4 below for two Rt series (April 2000 – December 2022): CHF/USD, purple line; and BLR/USD, green line. IFM-LN.76